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Vertex-Reinforced Random Walk on Z Has Finite Range

Robin Pemantle and Stanislav Volkov
The Annals of Probability
Vol. 27, No. 3 (Jul., 1999), pp. 1368-1388
Stable URL: http://www.jstor.org/stable/2652807
Page Count: 21
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Vertex-Reinforced Random Walk on Z Has Finite Range
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Abstract

A stochastic process called vertex-reinforced random walk (VRRW) is defined in Pemantle [Ann. Probab. 16 1229-1241]. We consider this process in the case where the underlying graph is an infinite chain (i.e., the one-dimensional integer lattice). We show that the range is almost surely finite, that at least five points are visited infinitely often almost surely and that with positive probability the range contains exactly five points. There are always points visited infinitely often but at a set of times of zero density, and we show that the number of visits to such a point to time n may be asymptotically nα for a dense set of values α ∈ (0, 1). The power law analysis relies on analysis of a related urn model.

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